Bell's theorem and the New Statistics.
Jay Sulzberger
On Thu, 4 Jul 1996, Ed Green wrote:
> Newsgroups: sci.physics
> Subject: Bell's Theorem and the New Statistics
>
> To the many people who have generously helped me with Bell's Theorem, and
> especially and most recently A. M. Steinberg and Keith Ramsay, my thanks.
> I hope they will not take it amiss if, before I am able to parse their
> detailed arguments, I drive a claim stake in the mud.
>
> I conjecture that it is impossible to make a good argument for non-locality
> from the type of data Bell's Theorem considers.
>
> Bear with me a moment, and consider two hypothetical experiments.
>
> Setup A involves the typical widely separated dectectors, apparently
> unable to influence each other by light speed signals (during the duration
> of a measurement event in their common lab frame). We find,
> hypothetically, that by flipping a toggle switch on detector 1 and later
> comparing the data from distant detector 2 that there was communication
> between the detectors, which propagated faster than light. In other
> words, the apparatus overall constitutes a superluminal signaling device.
>
> I don't think there is anybody who would doubt we had demonstrated
> superluminal signals or non-locality, then. I also don't think there is
> anybody knowledgeable who expects this to be the outcome of any real
> experiment.
>
> Setup B is precisely the same as setup A, except that now there is *no*
> signaling effect observed. No matter which way we flip the toggle, there
> is no way at detector 2 to deduce the position of the toggle (or just
> possibly, no way until a suitable lightspeed interval has passed). This
> is I believe what everybody expects, what seems to be observed, and is
> the kind of outcome from which we contemplate invoking Bell's reasoning to
> demonstrate non-locality.
>
> I contend there is *no* possible outcome from such an experiment that is
> incompatible with local propagation of effect; with a common cause for the
> correlated outcomes of widely separated detection events.
This is indeed the heart of the issue.
> I don't
> contend this is strictly obvious, but a naive inspection of the apparatus
> almost makes it so. Simply, the outgoing effect on either side of the
> source can be presumed to contain all the non-random aspects of the data
> stream at the detectors, with correlations of "paired" particles built in
> at the source.
Let C be circuit built of probabilistic gates having the following graph:
_____
| |
| |
|_____|
/ \
/ \
\ / \ /
\ / \ /
\_____ / \_____/
| | | |
| | | |
|_____| |_____|
| |
| |
| |
For any circuit built of (classical) probabilistic gates once we have
specified:
1. What is the set of signals for each wire.
2. What is the probabilistic behavior of each gate, including sources.
3. What is the signal we place on each wire whose tail is not
attached to a gate, i.e., what inputs we give the circuit.
Then:
4. We have determined the complete behavior of the whole circuit, including
what signals are on the internal wires. An internal wire is a wire whose
tail comes out of a gate, and whose head goes into a gate. This
determination includes also what signals are on the wires whose tails are
not attached to a gate, and those whose heads are not attached to any
gate. The input wires to the circuit we have directly from 3 above, and
the rest of the behavior we get by the 'propagation algorithm'.
Important: This total behavior is just a probability distribution,
because our gates are probabilistic, so we do not get, generally, any
sort of deterministic behavior.
5. Once we have the total behavior, we may just decide to look only at
the input/output behavior. We have the inputs and we have the
probabilities of any sentence of the form 'On wire 135 we have signal
alpha6, and on wire 337 we have no signal from the gamma or zeta signal
sets, and on wire 16 we have either signal alpha3 or a gamma signal.'
by 4 above. We now consider only sentences that do not use any wire
name that is the name of an internal wire. The resulting assignment of
probabilities, the I/O probabilities, with all input wire probabilities
actually just 0 or 1, is the I/O behavior at those inputs. Again this
is only a probability on a certain Boolean algebra of sentences, and
not a deterministic specification of which sentences are true or false,
though it could happen to be, of course, for various different reasons.
Now consider the set of all circuits of probabilistic gates with the above
circuit diagram, and whose left input wire has for set of signals
{ 0, 2*pi/3 }, and whose right input wire has for set of signals
{ 0, -2*pi/3 }. Further the left output wire has for set of signals
{ u, d }, and the right output wire has for set of signals
also { u, d}. We allow our circuits to have any arbitrary set of signals
on the remaining two internal wires, which wires are called the left
particle wire and the right particle wire. We also place no restrictions
on what the behavior of the top, or source, gate is, nor on the
other two gates, which are called the left measurement chamber gate, and
the right measurement chamber gate. Call this set of circuits the set of
Einstein-Podolsky-Rosen-Bohm circuits. Thus a member of this special set
of circuits is called an Einstein-Podolsky-Rosen-Bohm circuit.
Consider the following input-output behavior of a circuit with two inputs
called left and right, and two outputs called left and right. (We use a
standard notation, whose full explication is not easily found in the
literature, if one approaches these questions from the physicists side,
until recently, although long taught in introductory classes in circuit
theory, statistics, and partial differential equations.) Here is the
behavior:
p(LO = u and RO = u | LI = 0 and RI = 0) = 0
p(LO = u and RO = d | LI = 0 and RI = 0) = 1/2
p(LO = d and RO = u | LI = 0 and RI = 0) = 1/2
p(LO = d and RO = d | LI = 0 and RI = 0) = 0
p(LO = u and RO = u | LI = 0 and RI = -2*pi/3) = 3/8
p(LO = u and RO = d | LI = 0 and RI = -2*pi/3) = 1/8
p(LO = d and RO = u | LI = 0 and RI = -2*pi/3) = 1/8
p(LO = d and RO = d | LI = 0 and RI = -2*pi/3) = 3/8
p(LO = u and RO = u | LI = 2*pi/3 and RI = 0) = 3/8
p(LO = u and RO = d | LI = 2*pi/3 and RI = 0) = 1/8
p(LO = d and RO = u | LI = 2*pi/3 and RI = 0) = 1/8
p(LO = d and RO = d | LI = 2*pi/3 and RI = 0) = 3/8
p(LO = u and RO = u | LI = 2*pi/3 and RI = -2*pi/3) = 3/8
p(LO = u and RO = d | LI = 2*pi/3 and RI = -2*pi/3) = 1/8
p(LO = d and RO = u | LI = 2*pi/3 and RI = -2*pi/3) = 1/8
p(LO = d and RO = d | LI = 2*pi/3 and RI = -2*pi/3) = 3/8
We see how to read the above. The last line says that if we feed into the
left input wire the signal 2*pi/3, and feed into the right input wire the
signal -2*pi/3, then the probability of getting on the left output wire
the signal d and on the right output wire the signal d is 3/8. Similarly for
the rest of the lines. One may check that the the lines above do define a
probabilistic function from inputs to outputs. Let LI now stand for
the set of possible signals on the left input wire, RI stand for
the set of possible signals on the right input wire, LO to stand for the
set of possible signals on the right output wire, and RO to stand for
the set of possible signals on the right output wire. Thus our
probabilistic function, call it p, would be written p:LIxRI --> LOxRO ,
where the 'x' means Cartesian product.
Let us call the above probabilistic function p the Two Electron Singlet
Thirds function.
We may now state John Bell's Theorem.
Bell's Theorem. No Einstein-Podolsky-Rosen-Bohm circuit has as its input
output behavior the Two Electron Singlet Thirds function.
OK.
Note that p does not allow for signalling from the left to the right,
or vice versa. No matter what we put on the left input wire, we get the
same distribution of right output signals. One has to check this.
Since p, the Two Electron Singlet Thirds function does not allow for
signalling, I assume that you are claiming that you can indeed find an
find an Einstein-Podolsky-Rosen-Bohm circuit that has as its input output
behavior the Two Electron Singlet Thirds function.
> From an informational point of view this model seems very
> hard to reject.
>
> Bell in effect claimed this is incorrect, and that such discrimination is
> possible. I claim this must be a result of hidden or not so hidden
> ancilliary assumptions which go beyond the raw data. Not burdened with a
> prior model of the universe we would not be forced to reject the natural
> hypothesis that the distant arms of the experiment are not in
> communication. We must inject further ingredients from prior models (not
> simply statistical deduction based on the raw results) to conclude that.
There is a very important implicit mistaken theorem you are appealing to
here. The theorem has a true version. Feynman pointed it out in the
early eighties.
>
> I may be wrong. But as Lawrence Mead said though (roughly) "Strong
> claims require strong evidence". I have trouble seeing the possibility
> for any evidence at all here. Only signalling can force the rejection of
> the null hypothesis of non-communication. Other correlations may be built
> in causally from the source unless we further restrict the model with
> additional assumptions. Forget about what we "know" about physics for a
> moment, and just think about the data. We are fooling ourselves.
>
> At least that's the way it looks to me.
>
I hope the statement of Bell's Theorem as a statement of what a certain
class of circuits can compute is helpful. Of course, if we let our gates
be quantum gates, then yes, our circuit can compute p. Quantum locality
is different from classical locality. All quantum field theories are by
definition local, but they are quantum local. A quantum field theory is
just the infinitesimal version of a quantum circuit.
>
> --
>
> Ed Green / egr...@nyc.pipeline.com
>
> "Every one being allowed to learn to read, ruineth in the long run not
> only writing but also thinking." -- Nietzsche
>
I remain, as ever, your steady reader, Jay Sulzberger.
Edward Green
'ja...@panix.com (Jay Sulzberger)' wrote:
[Careful and explicit development]
>Bell's Theorem. No Einstein-Podolsky-Rosen-Bohm circuit has as its input
>output behavior the Two Electron Singlet Thirds function.
Even before I have understood your development, you leave very little
doubt you are stating a correct result, so this leaves me the task of
either:
(1) Finding what assumption in this model I believe is overly restrictive
in describing all possible "local" theories, or
(2) Finding the flaw in my own thinking.
>Note that p does not allow for signalling from the left to the right,
>or vice versa. No matter what we put on the left input wire, we get the
>same distribution of right output signals. One has to check this.
>Since p, the Two Electron Singlet Thirds function does not allow for
>signalling, I assume that you are claiming that you can indeed find an
>find an Einstein-Podolsky-Rosen-Bohm circuit that has as its input output
>behavior the Two Electron Singlet Thirds function.
>> From an informational point of view this model seems
very
>> hard to reject.
>>
>> Bell in effect claimed this is incorrect, and that such discrimination
is
>> possible. I claim this must be a result of hidden or not so hidden
>> ancilliary assumptions which go beyond the raw data. Not burdened with
>> a prior model of the universe we would not be forced to reject the
natural
>> hypothesis that the distant arms of the experiment are not in
>> communication. We must inject further ingredients from prior models
(not
>> simply statistical deduction based on the raw results) to conclude that.
>
>There is a very important implicit mistaken theorem you are appealing to
>here. The theorem has a true version. Feynman pointed it out in the
>early eighties.
Naturally you intrigue me. Perhaps this "hidden implicit theorem" I appeal
to will become clear in time. I have a very clear prior though imagining
packets of partial information about outcomes travelling in opposite
direction, correlated at birth and independently doing their things at the
detectors. I also have a very clear prior that allowing *any* effect of
adjustment of the left detector (left input channel) on the outcome at the
right, however statistically subtle, would in effect allow a practical
communication device to be built, and that in the absence of such effect
we cannot upset the correlated information packet model.
Well clearly something is wrong, and as I said, either (1) or (2) above
must apply. Naturally your post requires more thought, but this is my
first reaction. Thank you very much for posting this clear and powerful
model.
>I hope the statement of Bell's Theorem as a statement of what a certain
>class of circuits can compute is helpful. Of course, if we let our gates
>be quantum gates, then yes, our circuit can compute p. Quantum locality
>is different from classical locality. All quantum field theories are by
>definition local, but they are quantum local. A quantum field theory is
>just the infinitesimal version of a quantum circuit.
Now that is another very intriguing statement!
Jack Sarfatti
In <4rgm87$a...@news1.t1.usa.pipeline.com>
egr...@nyc.pipeline.com(Edward Green) writes:
>
>'ja...@panix.com (Jay Sulzberger)' wrote:
>
>[Careful and explicit development]
>
>>input
>>output behavior the Two Electron Singlet Thirds function.
>
>
What is the "Two Electron Singlet Thirds function" ? Sounds goofy. But
maybe I'm wrong. Please define it mathematically.
Edward Green
'sarf...@ix.netcom.com (Jack Sarfatti )' wrote:
>What is the "Two Electron Singlet Thirds function" ? Sounds
>goofy. But maybe I'm wrong. Please define it mathematically.
it without the development.
Edward Green
Two quick questions:
What explicitly is the propagation algorithm at a probabilistic gate?
How explicitly does a quantum gate differ from an ordinary probabilistic
gate?
Thanks.
Jack Sarfatti
In <4rjhce$a...@news1.t1.usa.pipeline.com>
You mean in the context of quantum computers? Oxford has a good web
page on this. Just use a search engine to find it.
Jack Sarfatti
In <4rhn30$d...@news1.t1.usa.pipeline.com>
original. You can do it by email.
Edward Green
Ok, I screwed up.
In penance, I will give a very simple example of a joint distribution
showing "non-locality". I am not actually sure that "distribution" is not
a misnomer, since part of the point seems to be that all the variables
involved *cannot* be described by a single joint distribution, unless
perhaps we make the choice of which set of variables to measure itself a
random variable. But that tends to suggest spurious side issues like
whether we are really free to choose which variable we measure, so let's
more neutrally call this data structure just that, a "data structure", or
a "behavior".
The setup is this: At two locations, Left and Right, we have the choice
of measuring one and only one datum, either A or B. Whichever one we
choose, the result is either "up" or "down". So a typical measurement
would be "LA = up" and "RA = down"; i.e., we chose channel A at the left
location, and the result was "up", etc.
For simplicity, we will assume all correlations are either +1 or -1. That
is, when "LA = up" it will either turn out RA is always up, or always
down.
Now, we will describe the observed correlations, and try to reproduce
them with a single distribution for all four variables: LA, LB, RA and RB.
To do this is equivalent to showing "locality"; because we could just
count instances of "up,up,down,down" and so forth, and then simulate the
results by sequentially send out the instructions "up,up" to the left and
"down,down" to the right. The detectors do not have to talk to each other
to decide what to do, they just read the tape. This is locality.
Here are the observed correlations:
(a) (LA,RA) = 1
(b) (LA,RB) = 1
(c) (LB,RA) = 1
(d) (LB,RB) = -1
Now, lets try to reproduce these as trials drawn from a single joint
distribution.
(a) implies that every instance for which LA = up also has RA = up. (b)
and (c) likewise imply that for each of these instances LB and RB will also
be up. But these instances now violate the requirment expressed in
(d)that RB be down when LB is up. End of story!
In short, these results cannot be expressed as the output of a single
joint distribution, but must in effect be regarded as resulting from
different distributions, somehow chosen when we select the pair of
variables to measure. Somehow the choice of channel A or B at Left must
be transmitted to the detector at Right, so it knows which distribution to
follow!
Nothing in this data structure permits human communication, though. Each
choice of distribution is consistent with observing, say, 50% up and 50%
down at each leg, which frequency we haven't even specified yet. So we
would have no way of knowing the setting of the other detector.
Nothing here is supposed to refer directly to any feature of quantum
mechanics. This is only supposed to illustrate very simply how an assumed
series of observations with "incompatiable" variables can be inconsistent
with the presumption of a series of partial trials from a single joint
distribution of all the variables, and hence with "locality". Once we
see this (once I see this), it becomes quite plausible that more subtle
effects are possible from the same kind of argument, regarding more
realistic data structures that might actually be calculated from quantum
mechanics.
[Nothing here has anything particularly to do with "determinism". The
observations at each detector are still bonafide random variables, and we
could even soften the correllations and reach similar conclusions more
difficultly.]
So I have nothing new to say, other than thinking that this way of looking
things -- that a data structure may be incompatible with description as a
series of partial observations from a common joint distribution, and hence
require internal communication of the experimenter's choice of measurent,
and that this is tantamount to "non-locality" -- may be a clearer statment
of affairs than I have heretofore encountered, and should you find it
novel or useful, I humbly ask you to remember where you read it.
[Note: Jacques Mallah points out to me that the *only* two assumptions
needed to derive Bell Inequality results are locality and "contrafactual
definitness". Without trying to disentangle them, I think we can see they
are both implied by the assumption of a single joint distribution.]
I thank all who helped me work this out, and especially Jay Sulzberger,
Jacques Mallah and Keith Ramsay.